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| Mirrors > Home > MPE Home > Th. List > fvopab4ndm | Structured version Visualization version GIF version | ||
| Description: Value of a function given by an ordered-pair class abstraction, outside of its domain. (Contributed by NM, 28-Mar-2008.) |
| Ref | Expression |
|---|---|
| fvopab4ndm.1 | ⊢ 𝐹 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} |
| Ref | Expression |
|---|---|
| fvopab4ndm | ⊢ (¬ 𝐵 ∈ 𝐴 → (𝐹‘𝐵) = ∅) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | fvopab4ndm.1 | . . . . 5 ⊢ 𝐹 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
| 2 | 1 | dmeqi 5915 | . . . 4 ⊢ dom 𝐹 = dom {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} |
| 3 | dmopabss 5929 | . . . 4 ⊢ dom {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐴 | |
| 4 | 2, 3 | eqsstri 4030 | . . 3 ⊢ dom 𝐹 ⊆ 𝐴 |
| 5 | 4 | sseli 3979 | . 2 ⊢ (𝐵 ∈ dom 𝐹 → 𝐵 ∈ 𝐴) |
| 6 | ndmfv 6941 | . 2 ⊢ (¬ 𝐵 ∈ dom 𝐹 → (𝐹‘𝐵) = ∅) | |
| 7 | 5, 6 | nsyl5 159 | 1 ⊢ (¬ 𝐵 ∈ 𝐴 → (𝐹‘𝐵) = ∅) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∅c0 4333 {copab 5205 dom cdm 5685 ‘cfv 6561 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-dm 5695 df-iota 6514 df-fv 6569 |
| This theorem is referenced by: fvmptndm 7047 |
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